AlexMennen's Shortform

8th Dec 20191 comment
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Theorem: Fuzzy beliefs (as in https://www.alignmentforum.org/posts/Ajcq9xWi2fmgn8RBJ/the-credit-assignment-problem#X6fFvAHkxCPmQYB6v ) form a continuous DCPO. (At least I'm pretty sure this is true. I've only given proof sketches so far)

The relevant definitions:

A fuzzy belief over a set is a concave function such that (where is the space of probability distributions on ). Fuzzy beliefs are partially ordered by . The inequalities reverse because we want to think of "more specific"/"less fuzzy" beliefs as "greater", and these are the functions with lower values; the most specific/least fuzzy beliefs are ordinary probability distributions, which are represented as the concave hull of the function assigning 1 to that probability distribution and 0 to all others; these should be the maximal fuzzy beliefs. Note that, because of the order-reversal, the supremum of a set of functions refers to their pointwise infimum.

A DCPO (directed-complete partial order) is a partial order in which every directed subset has a supremum.

In a DCPO, define to mean that for every directed set with , such that . A DCPO is continuous if for every , .

Lemma: Fuzzy beliefs are a DCPO.

Proof sketch: Given a directed set , is convex, and . Each of the sets in that intersection are non-empty, hence so are finite intersections of them since is directed, and hence so is the whole intersection since is compact.

Lemma: iff is contained in the interior of and for every such that , .

Proof sketch: If , then , so by compactness of and directedness of , there should be such that . Similarly, for each such that , there should be such that . By compactness, there should be some finite subset of such that any upper bound for all of them is at least .

Lemma: .

Proof: clear?